\(\bullet \)  Fractional Laplacian kernel. It is a truncated version of the fractional Laplacian kernel that has been widely studied in fractional and nonlocal diffusions (see e.g., [1]}, [2]}, [3]}, [4]}). We set
\(\phi _{true}(r) = c_{d,s} r^{-(d+2s)} \mathbf {1}_{[0.1,6]}(x) + 10^{d+2s} \mathbf {1}_{[0,0.1]}(x)\)  with exponent \(s = 0.5\)  and \(d = 1\) , where \(c_{d,s} = 4^s\pi ^{-d/2}\Gamma (d/2+s) \Gamma (-s)\) .
It is almost singular with multiscale values and its values near the singularity are crucial to the operator. It has \(R =6.51\) .
